I'm a Chinese. I hope this will help you:
请洗洗你的袜子 is an imperative sentence.
洗洗 (the same meaning of 洗一洗/洗一下) is a united verb. It's meaning is "wash". The same of “洗”
But, this wash (洗洗/洗一洗/洗一下) means "Hey, I need you to wash the socks." And “请洗你的袜子” is an ungrammatical sentence. Because 洗 always means "Okay, I'm washing them now." or "The socks are washed."
In addition, I would like to share you a little useful tip:
了/过 followed the verb means the event has been done and these two words can be combined. So “袜子已经洗过了” means "The socks have been already washed."
Sometimes we will say “快把袜子洗了”("Wash the socks quickly"). The 了 here have an implicit meaning of "the event will BE IN THE PAST". It's one of the forms of imperative sentence also.
有什么不懂的可以回复，我会尽量解答（If you have doubt about that, just comment below. I will help you as I can.）